Theoretical Compton profile of diamond, boron nitride and carbon nitride

Author’s Accepted Manuscript Theoretical Compton profile of diamond, boron nitride and carbon nitride Julio C. Aguiar, Carlos R. Quevedo, José M. Gomez, Héctor O. Di Rocco www.elsevier.com/locate/physb

PII: DOI: Reference:

S0921-4526(17)30405-2 http://dx.doi.org/10.1016/j.physb.2017.07.016 PHYSB310083

To appear in: Physica B: Physics of Condensed Matter Received date: 25 January 2017 Revised date: 24 May 2017 Accepted date: 7 July 2017 Cite this article as: Julio C. Aguiar, Carlos R. Quevedo, José M. Gomez and Héctor O. Di Rocco, Theoretical Compton profile of diamond, boron nitride and carbon nitride, Physica B: Physics of Condensed Matter, http://dx.doi.org/10.1016/j.physb.2017.07.016 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Theoretical Compton profile of diamond, boron nitride and carbon nitride Julio C. Aguiara,∗∗ a

Autoridad Regulatoria Nuclear, Av. Del Libertador 8250, (C1429BNP) Buenos Aires, Argentina.

Carlos R. Quevedob∗ b

Universidad Nacional de Asunci´ on, Facultad de Ciencias Exactas y Naturales (CC1039), Campus Universitario de San Lorenzo, Paraguay.

Jos´e M. Gomezb,c∗ c

Universidad Nacional de Asunci´ on, Facultad Polit´ ecnica (CC2111), Campus Universitario de San Lorenzo, Paraguay.

H´ector O. Di Roccod,e∗ d

Instituto de F´ısica “Arroyo Seco”, Facultad de Ciencias Exactas, Universidad Nacional del Centro de la Provincia de Buenos Aires, Pinto 399, (7000) Tandil, Argentina. e

Consejo Nacional de Investigaciones Cient´ıficas y T´ ecnicas (CONICET), Argentina.

Abstract In the present study, we used the generalized gradient approximation method to determine the electron wave functions and theoretical Compton profiles of the following super-hard materials: diamond, boron nitride (h-BN), and carbon nitride in its two known phases: βC3 N4 and gC3 N4 . In the case of diamond and h-BN, we compared our theoretical results with available experimental data. In addition, we used the Compton profile results to determine cohesive energies and found acceptable agreement with previous experiments. Keywords: Compton profile; generalized gradient approximation; diamond; boron and carbon nitride

∗ Corresponding

author Email address: [email protected] (Julio C. Aguiara,∗ )

Preprint submitted to Journal of LATEX Templates

July 7, 2017

1. Introduction The so-called super-hard materials, especially carbon, boron and nitrogen compounds, have been extensively studied. The structural bonding of these elements presents remarkable properties such as: high temperature resistance, wear-resistant coatings, compressibility, high thermal conductivity, and low density. These properties allow creating abrasive tools, cutting tools, electronic components, and optical parts [1, 2]. Among super-hard materials, diamond has the highest hardness and thermal conductivity. These properties determine greater industrial application in cutting and polishing tools. Gem diamond has a density of 3.513 g/cm3 and an atomic weight of 12.01 amu with face centered cubic (f cc) crystal structure where the lattice constant a = 3.5669 ˚ A in space group (227) and the Hermann– Mauguin notation is Fd3m [3]. Another hard material is boron nitride (h-BN), which has hexagonal form and a structure similar to that of graphite (space group (194) P63/mmc), where a = 2.504 ˚ A and c = 6.661 ˚ A, density = 2.18 g/cm3 , and atomic weight = 25.0124 amu [4]. Carbon nitride is also a hard material with has a structure equivalent to that of βSi3 N4 . In 1989, Liu and Cohen predicted the theoretical properties of βC3 N4 by using an empirical model and local-density approximation (LDA) [5, 6]. These authors showed that the bulk modulus of the hypothetical βC3 N4 solid (∼437 GPa) could have a hardness comparable to that of diamond (442446 GPa) with equivalent electronic densities. Other authors have predicted that the bandgap of βC3 N4 are high (3-4 eV) [7, 8, 9, 10, 11, 12, 13]. The polycrystalline hexagonal lattice of βC3 N4 (space group P3(143)) has been experimentally determined to be a = 6.36 ˚ A, and c = 4.648 ˚ A, whereas that of (C3 N4 in graphitic phase gC3 N4 ) (P6m2(187)) have been theoretically determined by using density-functional theory (DFT)-LDA, to be a = 4.74 ˚ A, and c = 6.72 ˚ A with a density of ∼2.3 g/cm3 [10, 14]. In the last decades, DFT has been widely used as a theoretical framework

2

to calculate the electronic structure of atoms, molecules and solids [15, 16]. On the other hand, LDA has provided a remarkably successful description for many electronic systems such as atomic, molecular and solid-state systems. However, in some applications where it is necessary to determine the wave functions, as in the case of Compton profile calculations, LDA shows that the profile is overestimated at low momentum transfer and underestimated around the Fermi momentum value. This failure has been attributed to the deficiency of LDA to describe the electronic exchange and correlation effects. A method that allows including the neglected exchange and correlation effects in LDA is the Lam–Platzman correction (LPC) [17, 18, 19]. This correction of the momentum density resolves the residual discrepancies outlined above and facilitates a critical assessment of Compton profiles. Recently, a semi-empirical method based on the joint use of the Fermi liquid model and Hartree–Fock formalism, has been proposed to include the effects of electron correlation [20]. In the present study, we were interested in the ability of the generalized gradient approximation (GGA) proposed by Perdew, Burke and Ernzerhof (PBE) [21] to describe the electron momentum density and isotropic Compton profile for diamond, h-BN, βC3 N4 and gC3 N4 by applying self-consistent solutions of the Schr¨ odinger–Kohn–Sham non–relativistic equations, under Troullier and Martins (TM) schemes [22]. The Compton profile allows verifying the quality of the electron wave functions through Fourier transforms, defining the Fermi surface and verifying the quality of band structure calculations [23, 24]. Compton profiles using these types of calculations have been previously reported for Be, Al, Ti, TiO2 and ZnO [25, 26, 27]. Diamond and h-BN have been widely studied both theoretically and experimentally by Compton profiles [28, 29, 30, 31, 32, 33, 34, 35]. In contrast, no Compton profile calculations have yet been reported for carbon nitride.

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2. Computational procedures One-electron wave functions for diamond, h-BN, βC3 N4 and gC3 N4 are calculated within the impulse approximation of the form ψnlm (r, θ, φ) = Rnl (r)Ylm (θ, φ),

(1)

where the atomic radial wave functions Rnl (r) for core electrons, the normconserving pseudo-potentials, and the pseudo-atomic orbitals for valence electrons are obtained using the ADPACK code [36]. The pseudo-potentials and orbitals are used as primitive basis input data for the execution of the OpenMx program (Open source package for Material eXplorer), which allows determining the Bloch wave functions of crystalline solids in the unit cell [37, 38, 39, 40]. The norm-conserving pseudo-potentials contain a charge density ρ (r) =  2 |Rnl (r)| r2 by orbital l constructed by standard schemes from all electron calculations. Assuming spherical screening, the Schr¨odinger–Kohn–Sham nonrelativistic equations under the GGA functional can be given by   l (l + 1) 1 ∂2 ef f + + Vl [ρ, r] rRnl (r) = rRnl (r) εnl . − 2 ∂r2 2r2

(2)

The effective potentials Vlef f can be decomposed into the local Hartree potential VH , the non-local exchange-correlation (XC) potential due to the valence and core electrons, and V ext , wich is the Coulomb potential of the nuclei:    ρ r ∂EXC [ρ] Vlef f [ρ, r] = + V ext (r) .  dr + |r − r | ∂ρ (r)

(3)

The ADPACK code allows including parameterizations of the LDA or GGA by using the exchange correlation energy  3 LDA [ρ] = ρ (r) εLDA EXC XC [ρ (r)] d r, where εLDA XC

3 [ρ (r)] = − 4

4

 1/3 3 ρ4/3 (r) . π

(4)

(5)

In the case of GGA, the exchange-correlation functional depends on the density and its gradient as follows:  GGA EXC [ρ] =

3

ρ (r) εGGA XC [ρ (r) , ∇ρ (r)] d r.

(6)

An improvement of the LDA functional is the GGA-PBE [21], represented by  3 GGA [ρ] = εLDA (7) EXC XC [rs (r)] FXC [rs (r) , s (r)] d r, where rs is the Wigner-Seitz radius and is related to Fermi’s momentum rs = 1.9192/kF . The function FXC is given by FXC [rs , s] = FX [s] + FC [rs , s] where S=

(8)

|∇ρ| , 2ρkF

(9)

and FX [s] = 0.804 −

0.804 1+

0.21951 2 . 0.804 s

(10)

2.1. Compton profile The isotropic Compton profile Jnl (q) for the nl orbital is defined as the projection of the electron momentum density ρ(p) along the scattering vector after the collision

Jnl (q) =

1 2



∞ |q|



2

|χnl (p)| pdp =

1 2



∞

ρ(p) pdp,

(11)

|q|

where χnl (p) is called momentum space wave function and can be obtained via the Fourier transforms of Rnl (r) according to  1/2  ∞ 2 l χnl (p) = (−i) Rnl (r) jl (pr) r2 dr, π 0

(12)

where jl (pr) is the spherical Bessel function of the first kind, p is the momentum of the electron in the atom, and q is the projection of the momentum transfer k on the electron momentum p before collision: q=−

5

k.p . k

(13)

2.2. Cohesive energy by Compton profile The total ground-state energy ET of electrons in crystals can be obtained from the solution to the Schr¨odinger equation in momentum space [41]. The expectation value of kinetic energy Ek  ≡ p2 /2 follows from the application of the virial theorem, which connects the expectation values of Ek  and potential V  energy of a system as follows: Ek  = − ET  = − V  /2.

(14)

The application of the virial theorem allows determining the Ek  of an isotropic system via  3 ∞

Jnl (q)q 2 dq. (15) Ek  = 2 0 Additionally, it is possible to determine the cohesive energy from the differences between calculated free-atomic J(q)atom and J(q)crystal in crystalline solids. Thus, cohesive energy Ecoh/atom is defined as the difference between the total ground-state energy of the Compton profile of the crystal and the sum of the individual atoms, as follows: Ecoh/atom

1 = 2



∞ 0

[J(q)crystal − J(q)atom ] q 2 dq.

(16)

The values of J(q)atom can be obtained from Hartree-Fock Compton profile calculations [42], and the values of J(q)crystal can be measured or calculated. In the present case, the values of J(q)crystal are obtained from the present GGAPBE-TM scheme and those of J(q)crystal are determined via Hartree-Fock wave functions by using Fischer’s code [43]. The application of this method has the advantage that at high values of q, where the contribution is limited to the core electrons, the integral is small, whereas at low values of q, the integral is significant. 3. Results and discussion In all cases, the valence states were expanded over s–, p–, and d–like character where the valence electron densities distribution obtained from tabulated 6

radial GGA-PBE wave functions are shown in Figure (1). The Monkhorst-Pack k-point grid used were 3x3x3 for diamond, 20x20x1 for h-BN, 3x3x8 for βC3 N4 and 6 x 6 x 4 for gC3 N4 . To facilitate comparison between the different structures, the radial valence electron density was normalized to units according to 

∞

ρ (r) = 0



GGA (r) Rnl

2

r2 dr.

(17)

Figure 1: The color online: Radial valence electron density; the black solid curve denotes diamond, the solid blue squares denote h-BN; and the red solid circles denote βC3 N4 . Inset: values of the densities in the ranging from 3 to 5 a.u.

In the case of diamond, the presence of one prominent hump of 1.62 at 1.10 a.u. of the center and another one 3.9 a.u is the result of the strongly attractive C 2p potential of the purely covalent diamond. In h-BN, the valence electron density is strongly localized near the N and exhibits only a single peak of height 1.60 at 0.96 a.u. In the case of βC3 N4 a hump of 1.50 at 0.99 of the center is observed. Thus, the heteropolarity of the C-N bond in βC3 N4 lies intermediate between purely covalent diamond and partially ionic boron nitride. The theoretical Compton profiles (Table 1) were compared with previous experiments for diamond Figure (2) [29] and hBN Figure (3) [31]. The differences between βC3 N4 and gC3 N4 are shown in Figure (4). 3.1. Diamond We assumed that the solid bulk has a face-centered cubic structure, where by minimization of the energy, the lattice constant is determined to be a = 3.4969

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Table 1: Isotropic Compton profile calculations for diamond, h-BN, βC 3 N4 and gC3 N4 obtained by using GGA-PBE schemes normalized to 2.90, 5.86, 6.34 electrons equal to the area under 0 to + 7 a.u., respectively.

q(a.u) 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5

diamond 2.0667 2.0581 2.0321 1.9887 1.9271 1.8490 1.7538 1.6437 1.5208 1.3886 1.2513 1.1150 0.9432 0.7811 0.6450 0.5353

h-BN 4.4339 4.4180 4.3629 4.2511 4.0655 3.8122 3.5015 3.1616 2.8212 2.5003 2.2110 1.9531 1.7262 1.5258 1.3492 1.1936

βC3 N4 4.7378 4.7032 4.6023 4.4395 4.2232 3.9634 3.6736 3.3647 3.0475 2.7373 2.4397 2.1614 1.9086 1.6826 1.4843 1.3125

gC3 N4 4.7490 4.7140 4.6121 4.4477 4.2294 3.9673 3.6752 3.3642 3.0452 2.7336 2.4352 2.1565 1.9038 1.6782 1.4804 1.3093

q(a.u) 1.6 1.7 1.8 1.9 2.0 2.2 2.4 2.6 2.8 3.0 3.5 4.0 4.5 5.0 6.0 7.0

diamond 0.4590 0.4005 0.3562 0.3224 0.2964 0.2584 0.2313 0.2103 0.1937 0.1796 0.1497 0.1203 0.0903 0.0640 0.0338 0.0210

h-BN 1.0592 0.9427 0.8431 0.7579 0.6872 0.5740 0.4906 0.4225 0.3688 0.3233 0.2373 0.1768 0.1321 0.0989 0.0559 0.0315

˚ A and the experimental value is a = 3.5669 ˚ A. The Brillouin zone determined by k-point sampling grids on super cell size was 3x3x3. The cohesive energy of diamond computed via the theoretical Compton profile data using Eq.(16) is -0.2612 a.u./atom, being the experimental value -0.271 a.u./atom [44]. The difference between the experimental Compton profile for diamond and our calculations is shown in Fig.(2). 3.2. h-BN The hexagonal boron nitride form of h-BN was determined as a structure similar to that of graphite, where the lattice constants determined by minimization of the energy are a =2.5528 ˚ A and c =6.7905 ˚ A, and where experimental values are a = 2.504 ˚ A and c = 6.661 ˚ A. The cohesive energy of diamond was computed via the theoretical Compton profile, where the value found was -0.2379 a.u./atom, being the experimental value -0.2425 a.u./atom [45]. The difference between the experimental Compton profiles for h-BN and our calculations is shown in Figure (3). 8

βC3 N4 1.1653 1.0401 0.9334 0.8427 0.7652 0.6393 0.5453 0.4698 0.4103 0.3602 0.2669 0.2010 0.1521 0.1150 0.0664 0.0377

gC3 N4 1.1627 1.0379 0.9318 0.8415 0.7643 0.6399 0.5458 0.4702 0.4105 0.3602 0.2670 0.2011 0.1521 0.1150 0.0664 0.0377

Figure 2: Empty circles denote differences between experimental Compton profiles of Reed and Eisenberger [29] and our GGA-PBE calculations [ΔJ = J exp − J theo ]. The size of the circles corresponds to the experimental resolution, which is less than 0.4 a.u.

Figure 3: Squares denote differences in the Compton profiles between our approach (GGAPBE) and previous experimental results for h-BN [35]. Circles denote our approximation and previous experimental results [31].

3.3. βC3 N4 and gC3 N4 The polycrystalline hexagonal lattice of βC3 N4 has been theoretically determined to be a = 6.3937 ˚ A and c = 4.4011 ˚ A, whereas for gC3 N4 the equilibrium structural parameters theoretically determined using GGA-PBE were a = 4.7458 ˚ A and c = 6.7258 ˚ A. The Cohesive energy of βC3 N4 was computed via the theoretical Compton profile, where the value found was -0.2076 a.u./atom, being -0.2131 a.u./atom the value predicted by Liu and Cohen [5, 6]. the differences between our theoretical Compton profiles for βC3 N4 and gC3 N4 are shown in Figure (4). The differences observed between the Compton profiles of βC3 N4 and gC3 N4 show that the electron momentum density distributions have different electronic

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Figure 4: Circles denote the differences in the Compton profiles between gC3 N4 and βC3 N4 .

behavior, probably attributable to the strong anitropy of the two phases. 4. Conclusions The structural and electronic properties of diamond, hexagonal boron nitride (h-BN) and carbon nitride in its two known phases (βC3 N4 and gC3 N4 ) were calculated from all-electron wave functions. Good agreement between cohesive energies and Compton profile experiments was observed in all cases. With respect to the structural properties of diamond, we were able to reproduce the experimental values for the cohesive energy within 3 %. This proves the excellent capability of the generalized gradient approximation to calculate the electronic properties of covalent such as diamond and semi-ionic materials (BN and C3 N4 ). In addition, we demonstrated that the Compton profile of diamond can be estimated with relative accuracy (1-5%) when compared to experimental results. However, the results obtained for h-BN were not good when compared with previous experiments performed by Tyk et al. [35], but showed excellent agreement with those performed by h-BN Ahuja et al. [31]. The valence electrons contribute to the Compton profile with a sharp peak located at low momentum transfer, it is possible to compare the behavior of the materials by comparing the electron momentum density distribution. Based on wave functions calculations and electron momentum density of this covalent solids we can concluded that h-BN and Carbon Nitride have a similar behavior in contradistinction to the diamond (see Fig.1). In summary, our study of the electronic properties of 10

diamond, boron nitride and carbon nitride via the Compton profile offers additional information of these materials, such as the fact that βC3 N4 and gC3 N4 show small differences, probably attributable to the strong anitropy of the two phases. Acknowledgments: We thank the support from Universidad Nacional del Centro de la Provincia de Buenos Aires (UNCPBA, Argentina), the Universidad Nacional de Asunci´on (UNA-FaCEN, Paraguay) and the Nuclear Regulatory Authority of Argentina.

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